Chord length ($L=2r\sin (\theta /2)$)

For a circle of radius $r$, a chord with minor central angle $0\le \theta \le \pi$ has lengthBisecting the isosceles triangle formed by its two radii gives a right triangle with hypotenuse $r$ and opposite side $L/2$, proving the formula.

If the perpendicular distance from the center to the chord is $d$, the same right triangle gives $L=2\sqrt{r^2-d^2}$. For $r=5,d=3$, the length is eight. The maximum chord is the diameter $2r$, obtained at $d=0$ or $\theta =\pi$.

The corresponding minor arc has length $r\theta$ when the angle is measured in radians. For small angles the chord is approximately that arc length, since $\sin (\theta /2)\sim \theta /2$, but for a nonzero angle the straight chord is strictly shorter than the arc.